ida2 - Metamath Proof Explorer

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Theorem ida2 17302

Description: Morphism part of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)

**Hypotheses**
Ref	Expression
idafval.i	⊢ 𝐼 = (Id_a‘𝐶)
idafval.b	⊢ 𝐵 = (Base‘𝐶)
idafval.c	⊢ (𝜑 → 𝐶 ∈ Cat)
idafval.1	⊢ 1 = (Id‘𝐶)
idaval.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)

**Assertion**
Ref	Expression
ida2	⊢ (𝜑 → (2^nd ‘(𝐼‘𝑋)) = ( 1 ‘𝑋))

**Proof of Theorem ida2**
Step	Hyp	Ref	Expression
1		idafval.i	. . . 4 ⊢ 𝐼 = (Id_a‘𝐶)
2		idafval.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
3		idafval.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
4		idafval.1	. . . 4 ⊢ 1 = (Id‘𝐶)
5		idaval.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
6	1, 2, 3, 4, 5	idaval 17301	. . 3 ⊢ (𝜑 → (𝐼‘𝑋) = ⟨𝑋, 𝑋, ( 1 ‘𝑋)⟩)
7	6	fveq2d 6660	. 2 ⊢ (𝜑 → (2^nd ‘(𝐼‘𝑋)) = (2^nd ‘⟨𝑋, 𝑋, ( 1 ‘𝑋)⟩))
8		fvex 6669	. . 3 ⊢ ( 1 ‘𝑋) ∈ V
9		ot3rdg 7691	. . 3 ⊢ (( 1 ‘𝑋) ∈ V → (2^nd ‘⟨𝑋, 𝑋, ( 1 ‘𝑋)⟩) = ( 1 ‘𝑋))
10	8, 9	ax-mp 5	. 2 ⊢ (2^nd ‘⟨𝑋, 𝑋, ( 1 ‘𝑋)⟩) = ( 1 ‘𝑋)
11	7, 10	syl6eq 2872	1 ⊢ (𝜑 → (2^nd ‘(𝐼‘𝑋)) = ( 1 ‘𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3486 ⟨cotp 4561 ‘cfv 6341 2^nd c2nd 7674 Basecbs 16466 Catccat 16918 Idccid 16919 Id_acida 17296

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-sn 4554 df-pr 4556 df-op 4560 df-ot 4562 df-uni 4825 df-iun 4907 df-br 5053 df-opab 5115 df-mpt 5133 df-id 5446 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-2nd 7676 df-ida 17298

This theorem is referenced by: arwlid 17315 arwrid 17316

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